Solve The Following Inequality Algebraically Calculator

This guide explains how to solve inequalities algebraically using our calculator. Whether you're dealing with linear, quadratic, or polynomial inequalities, we'll walk you through the process step by step.

Introduction

Inequalities are mathematical statements that compare two expressions using symbols like <, >, ≤, or ≥. Solving inequalities involves finding all values of the variable that make the inequality true. This guide will help you solve inequalities algebraically with our calculator.

Our calculator handles linear, quadratic, and polynomial inequalities. It provides step-by-step solutions and visual representations of the solution sets.

Basic Steps to Solve Inequalities

Identify the type of inequality (linear, quadratic, etc.)
Isolate the variable term on one side of the inequality
Perform the same operation on both sides to maintain the inequality
Consider the direction of the inequality when multiplying or dividing by negative numbers
Write the solution in interval notation or describe the solution set

Key Formula

For a linear inequality like ax + b > c, the solution is found by solving ax + b = c and then determining the direction of the inequality based on the value of a.

Types of Inequalities

Linear Inequalities

Linear inequalities have the form ax + b > c, ax + b < c, etc. They can be solved by isolating the variable term.

Quadratic Inequalities

Quadratic inequalities have the form ax² + bx + c > 0. They require finding the roots of the corresponding equation and testing intervals.

Polynomial Inequalities

Polynomial inequalities involve higher-degree polynomials. The solution process is similar to quadratic inequalities but requires more complex analysis.

Worked Examples

Example 1: Linear Inequality

Solve 3x - 5 > 10.

Add 5 to both sides: 3x > 15
Divide by 3: x > 5

Solution: x ∈ (5, ∞)

Example 2: Quadratic Inequality

Solve x² - 4x - 5 > 0.

Find roots: x = 5 and x = -1
Test intervals: (-∞, -1), (-1, 5), (5, ∞)
Solution: x ∈ (-∞, -1) ∪ (5, ∞)

Common Mistakes

Forgetting to reverse the inequality sign when multiplying or dividing by a negative number
Incorrectly identifying the critical points for quadratic inequalities
Miscounting the number of roots when solving polynomial inequalities

Tip

Always double-check your work and consider testing values in the solution set to verify your answer.

FAQ

What is the difference between an equation and an inequality?

An equation states that two expressions are equal (using =), while an inequality states that one expression is greater than, less than, or not equal to another (using <, >, ≤, ≥, or ≠).

How do I solve compound inequalities?

Compound inequalities are solved by finding the intersection of the solutions to each part of the inequality. For example, to solve 1 < x < 5, you would solve each part separately and find the overlapping range.

What is interval notation?

Interval notation is a way to represent sets of real numbers using parentheses and brackets. Parentheses ( ) indicate that an endpoint is not included, while brackets [ ] indicate that an endpoint is included.